A question about Inner and outer Automorphism

Question

Suppose G is a not abelian group but finite, H$\unlhd$G and K$<$G with H$\bigcap$K=1, then to any k$\in$K, $\phi_k$(h)=$khk^{-1}$ is $\in$Aut(H), my question is: is that possible for some $\phi_k$ $\in$ Inner Automorphism of H and not trivial? If possible, could you give me an example? Thank you!

Answer 1

As an example, let $X$ be any nonabelian group, $G = X \times X$, $H = \{(x,1) \mid x \in X \}$ and $K = \{(x,x) \mid x \in X \}$.

Then, for $k = (x,x) \in K$, $\phi_{k}$ acts on $H$ as conjugation by $(x,1) \in H$.