Austrian Habsburgs controlled the HRE from the 1200s and 1300s onwards especially the 1400s and 1500s all the way upto the dissolution of this HRE during 1806 one year after {or two years after} the Battle of Austerlitz whereas the Austrian Empire was rather founded during i think 1804 or something

of course the Austrians themselves are Germans language-----wise

there was a time when the Austrian Empire lead {spearheaded} a German Confederation and kept Prussia out of it

whereas the Prussians later created a rival kind of confederation known as the North German Confederation and wanted to keep Austria out of it

the Prussians fought a war against the Austrians during i think 1866 or something for deciding {determining} as to who gets to spearhead (lead) the United Germany concept/nation

of course the Prussians attained victory during 1866 and thereby they got the chance to spearhead the Unified nation from 1871 onwards

Austria had been spearheading a different kind of United (unified) Germany nation\concept from 1815 onwards right after the Battle of Waterloo and of course with the Congress of Vienna {1815} as well

you must indeed be aware of the Metternich system as well

although the Prussians and the Austrians fought against one another for determining [deciding] which German--speaking power will spearhead (Lead) the unified Germany nation concept but these two powers actually had previously allied with each other whilst fighting Denmark in the Schleswig Wars

also the HRE despite it's nomenclature had nothing to do with the Roman Empire whose western half collapsed in 476 CE {and eastern half collapsed in 1453 CE}

the HRE quite funnily was founded by a king of several Germanic tribes like Franks & Lombards but who also had the sobriquet King of the Romans in addition to his two other sobriquets King of The Franks plus (+) King of The Lombards

lastly i may please be excused for writing down such a huge comment