Cause the inverse is strictly related to the function itself, that especially gives $p^{-1}(y\tilde{y})=p^{-1}(p(x)p(\tilde{x}))=p^{-1}(p(x\tilde{x}))=x\tilde{x}=p^{-1}(y)p^{-1}(\tilde{y})$.
But this relation is really just a "happy accident". Most properties won't be heridated e.g. the inverse of a continuous function is not necesarily continuous or e.g. differentiability.
Moreover, I'd like to stress that -despite the fact that most textbook define isomorphism of groups to be bijective homomorphism- what one really desires is a homomorphism which inverse is homomorphism as well, what luckily comes for free ;-)