Let $G$ be a finite group with $M$ a specified subgroup. One way of constructing representations of $G$ from $M$ is by induction, however if the index of $M$ in $G$ is large, then the composition series of induced representations can be tricky to determine.
If one could instead find an intermediary subgroup $M \leq P \leq G$ such that $M$ is isomorphic to a quotient of $P$ then we could lift our representation from $M$ to $P$, before inducing from $P$ to $G$, thus reducing the index by a factor of $\left| P : M \right|$.
In such a case, $P$ is a split extension of $M$ in $G$, and so if $M \cong P/U$ for some normal subgroup $U \leq P$, then $P$ is the internal semi-direct product of $U$ and $M$ in $G$. Thus determining such a $P$ is equivalent to choosing a subgroup $U$ of $G$ with $U \cap M = \left\{ e \right\}$.
Then since $P = MU$, the factor by which we reduce is $\left| MU :U \right| = \left| U \right|$, and so we are left with the problem of determining a subgroup $U$ of $G$ such that $U \cap M = \left\{e\right\}$ and that maximises $\left| U \right|$.
Is there any systematic way of determining such a $U$?
Edit: I think we also have to add the restrictions that $MU$ is a subgroup of $G$, and that $U$ is normal in $MU$.