  Given a finitely presented group $G$, we wish to explore the conditions under
which automorphisms of quotients $G/N$ can be lifted to automorphisms of $G$.
We discover that in the case where $N$ is a central subgroup of $G$, the
question of lifting can be reduced to solving a certain matrix equation. We
then use the techniques developed to show that $Inn(G)$ is not characteristic
in $Aut(G)$, where $G$ is a metacyclic group of order $p^n$, $p\neq 2$.
