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.

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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 $$\phi(C_G(H)) = C_K(\phi(H))$$
In the case of automorphism we don't have that the centralizer is preserved because not all subgroups are preserved right? i.e. we don't have $\phi(H)=H$ for all subgroups $H$ – user150391 Aug 24 '14 at 15:09
Yeah, $C_G(\phi(H)) \neq C_G(H)$ in general, but that's because you failed to relabel totally. The correct statement is $\phi(C_G(H)) = C_G(\phi(H))$, so "centralizer of image is image of centralizer". Unless you know $H$ is a characteristic subgroup i don't know how you could hope to say more. – JHance Aug 24 '14 at 15:12