Let $H$ be a group, $\phi$ an automorphism of $H$ of order n and fix $h_0 \in H$. I wonder, what the restrictions are, such that $$G:= \lt H,g \mid g^n=h_0,\quad \forall h \in H: ghg^{-1}=\phi(h) \gt$$ defines a group which has $H$ as normal subgroup such that $G/H$ is cyclic of order n.

There are two obvious restrictions:

(1) $h_0$ has to be in the center of $H$. For, $h_0hh_0^{-1} = g^nhg^{-n}=\phi^n(h) = h$, since $\phi$ has order n.

(2) $\phi(h_0) = h_0$. For, $\phi(h_0) = gh_0g^{-1}=gg^ng^{-1} = g^n = h_0$.

But I can't figure out, if there some more restrictions.

I tried to apply the classification of extensions with non-abelian kernel (Kenneth Brown, Cohomology of Groups, chapter IV, §6), but that requires to consider $H^3(Out(H),C)$ ($C$ the center of $H$) and I'm unable to do so, because I have no information about $Out(H)$ and $C$.

Any help is appreciated.