The fossil and archaeological records, as well as ethnographic studies of modern-day hunter-gatherers, indicate that women have a long history of hunting game.
Unfortunately, both these word sets assume a binary, which does not exist biologically, psychologically or socially. Sex and gender both exist as a spectrum, but when citing the work of others, it is difficult to add that nuance.
Overall, females are metabolically better suited for endurance activities, whereas males excel at short, powerful burst-type activities. You can think of it as marathoners (females) versus powerlifters (males). Much of this difference seems to be driven by the powers of the hormone estrogen.
Their skeletons show the same patterns of wear and tear. This finding suggests that they were doing the same things, from ambush-hunting large game animals to processing hides for leather. Yes, Neandertal women were spearing woolly rhinoceroses, and Neandertal men were making clothing.
Individuals get injured or die, and the availability of animal and plant foods changes with the seasons. All group members need to be able to step into any role depending on the situation, whether that role is hunter or breeding partner.