Image of centralizer under an isomorphism Suppose we have a group isomorphism $\phi: G\rightarrow K$ between two finite groups and let $H$ a subgroup of $G$. Are there any known facts about the image of the centralizer $C_G(H)$ of $H$ in $G$ under $\phi$? 
Also, same question for the special case of $K=G$ i.e. when $\phi$ is an automorphism.

Morally, you can view isomorphism as just relabeling the elements of a group without changing any structure. So the best we can hope to say is that $$\phi(C_G(H)) = C_K(\phi(H))$$
Edit: I realize now that "best we can hope to say" is a bit strange. Basically any structure you define in group theoretic terms will carry over precisely under isomorphism. 
