  For each integer $k \geq 2$ or $k=\infty$, one can associate to each Riemann
surface $S$ its $k$-homoly cover $\widetilde{S}_{k}$. This surface admits a
group of conformal automorphisms $M_{k} \cong {\rm H}_{1}(S;{\mathbb Z}_{k})$,
where ${\mathbb Z}_{\infty}={\mathbb Z}$, such that
$S=\widetilde{S}_{k}/M_{k}$. If $L \leq {\rm Aut}(S)$, then there is a short
exact sequence $1 \to M_{k} \to \widetilde{L}_{k} \to L \to 1$, where
$\widetilde{L}_{k}$ is a subgroup of conformal automorphisms of
$\widetilde{S}_{k}$.
  If $k \neq \infty$ and $S$ is analytically finite and non-exceptional, then
$M_{k}$, $\widetilde{L}_{k}$ and $L$ are finite groups. In this case, if $k$
and $|L|$, the order $|L|$ of $L$ are relatively prime integers, then the
Schur-Zassenhaus theorem asserts that $\widetilde{L}_{k} \cong M_{k} \rtimes
L$.
  In this paper, we investigate situations when $k$ and $|L|$ are not
necessarily relatively primes.
