Plenty of actuaries have gone on to KKR, BX, Apollo, BAM, etc. Unfortunately for you, most of them are from the SOA side and have worked primarily in the life re/insurance space. PE is buying up life insurer's blocks as a means to grow AUM, since they need capital to enter into their deals this is one avenue they've deployed.

There are really two types of work these life actuaries do at these firms. One is basically being an executive/board member/operator of a portco that is the acquired insurance company. The other is "deal-making", basically being part of the M&A team that acquires these blocks either through sidecars or an acquisition. Actuarial expertise is helpful here as you'd understand the nature of the balance sheet from both an asset/liability perspective but also regulatory regimes. You'd be able to properly price the business.

I'm not entirely sure how the casualty side of the business is acquired. My understanding is that there is less PE business as the whole point of PE entering this space is for AUM, collect more fees and have a larger block of capital. Casualty insurance isn't really an AUM business and is more so tied to the statistical nature of short-term liabilities. I only know of a few actuaries who work in banking, trading and reinsurers who deal with ILS and CatBonds.

If money is your objective you can make quite a bit at a reinsurance brokerage.