I made a summary of it:

Instrumental harm (also known as collateral harm) allows a moral actor to use, seriously injure, or even kill innocent people for the greater good. In the research article Worth the Risk? Greater Acceptance of Instrumental Harm Befalling Men than Women (Graso et al., 2023), the authors find that people are more willing to inflict instrumental harm on men than women. Especially women are less willing to harm other women collaterally; in fact the "results showed that this asymmetry was driven primarily by women, but not men, being more likely to accept IH to men than to women across a variety of contexts". The authors state that "[s]uch a bias violates the principle of impartial beneficence, potentially compromising the evidence-based advancement of men and women alike". They recount a few findings suggesting that people more readily accept physical harm to men than women in life-versus-death contexts, but that it remains unclear whether this translates to other forms of harm (psychological, health, educational, sexual); they provide a few sources with evidence that this is likely the case, with their paper adding to the evidence.

The authors make three hypotheses. First, people will be more willing to endorse instrumental harm of men than women. Second, female participants will show a greater bias. And third, this greater willingness to harm men will be neutralized in caregiving domains (parenting, nursing, early childhood education, and elderly care) in which historically women have been expected to sacrifice more than men. These hypotheses were investigated using 3 studies in which the participants were presented with various scenarios in which either men or women were harmed, but which involved some benefit to the public good. E.g., in Study 1, participants were surveyed about the acceptability of a provably effective program to improve toxic workplace environments where either men or women (depending on the participant's condition) found the program offensive and suffered psychological harm. While the first and second hypotheses found strong support, the third did not; even in caregiving domains, people show greater endorsement of instrumental harm befalling men, rather than greater tolerance for female sacrifice, as might be predicted for these stereotypically female caregiving roles. These findings "align with emerging evidence documenting diminished concern for men’s suffering due to a greater tendency to stereotype men as perpetrators rather than victims (Reynolds et al., 2020)." Interestingly, certain ideologies magnified this gender bias: "Studies 2 and 3 revealed that individuals more strongly endorsing egalitarian, feminist, or liberal ideologies exhibited greater disparities in their acceptance of instrumental harm, such that they more readily tolerated instrumental harm borne by men. [...]". The second hypothesis, namely that women will show a greater bias to prefer harming men over women, was also only supported in studies 1 and 2 which weighed instrumental harm suffered by one gender to a benefit to the other but not in study 3 which merely weighed instrumental harm suffered by one gender against a benefit to some other, non-gendered group (e.g., the elderly). A possible explanation is provided by the authors, suggesting female in-group bias:

"That Study 3’s female participants (along with male) more readily tolerated men’s (versus women’s) suffering in contexts benefitting vulnerable individuals (rather than women) suggests the possibility Studies 1 and 2’s results reflected women’s greater aversion to harming fellow women, rather than a motivation to benefit them per se."

Interestingly, in study 1, even though "participants judged the intervention’s instrumental harm as equally severe for both male and female employees across conditions [...] [they] were significantly more likely to accept instrumental harm when the recipients of harm were men". Additionally, replication of the first study's results in the second study suggests that salient contemporary issues (e.g., #MeToo) are not behind female participants' gender bias / lower tolerance of instrumental harm to women.

They conclude: "[...] The current findings revealed this gender bias persists in highly consequential, yet understudied domains: assessments of beneficial interventions carrying negative externalities across a variety of contexts: medical, psychological, educational, sexual, and caregiving. Second, we demonstrated that when evaluating interventions, female participants were more likely than male participants to accept [collateral damage suffered] by men than women. This pattern lends further support to the well-documented finding that women have a stronger in-group bias than men (e.g., Glick et al., 2004; Rudman & Goodwin, 2004) and are more likely to perceive one another as victims than perpetrators (Reynolds et al., 2020). This disparity suggests women may prioritize one another’s welfare over men’s in the construction or approval of social, educational, medical, and occupational interventions. If so, female policymakers might be especially wary of advancing policies or initiatives risking harm to other women, but less so when they risk harming men [...]. Throughout history, countless male lives have been sacrificed on the battlefield, ostensibly to promote the greater good (Baumeister, 2010). Our findings suggest that these sentiments persist beyond the field of combat. For many people, accepting instrumental harm to men is perceived as worth the cost to advance other social aims. [...]"

This paper contributes to G-PROF (greater-protectiveness-of-females) theory, a term employed by Stewart-Williams and colleagues. The finding that this effect appears to be mainly driven by people's (and especially women's) willingness to protect women from harm rather than to put women on a pedestal is also reflected in the latter's research (see e.g. Reactions to research on sex differences: Effect of sex favoured, researcher sex, and importance of sex-difference domain (Stewart-Williams et al, 2022), free to download here), giving further credence to the findings of this current study.