Let me answer this question under the assumption that we are talking about `isomorphism' of groups. So I have a group $G$ containing two isomorphic subgroups $H$ and $K$. When can I extend the isomorphism between $H$ and $K$ so that it becomes an automorphism of $G$? The answer is that **usually you can't**.

In particular the model theoretic concept of *homogeneity* is relevant here: roughly speaking (because I am no model theorist) a class of structures is homogeneous if whenever you have three structures $H$, $K$ and $G$ in the class such that (a) $H$ and $K$ are substructures of $G$ and (b) $H$ is isomorphic to $K$, then an isomorphism between them can be extended to an automorphism of $G$.

An example of a class of groups that is homogeneous would be $\mathcal{E}_p$, the class of elementary abelian $p$-groups. That the isomorphism is always extendible follows from the fact that any linearly independent set of vectors in a vector space can be extended to a basis.

To return to the original question: in addition to Bruno's example given above, here is another: $G=C_2\times C_4$ and $H$ and $K$ are cyclic subgroups of order $2$. Take $H$ to lie in a subgroup $C_4$ and $K$ to not lie in such a subgroup. Note that $G$ is abelian so $H$ and $K$ are normal, which deals with your second question.