  L. Babai has shown that a faithful permutation representation of a nonsplit
extension of a group by an alternating group $A_k$ must have degree at least
$k^2(\frac{1}{2}-o(1))$, and has asked how sharp this lower bound is. We prove
that Babai's bound is sharp (up to a constant factor), by showing that there
are such nonsplit extensions that have faithful permutation representations of
degree $\frac{3}{2}k(k-1)$. We also reprove Babai's quadratic lower bound with
the constant $\frac{1}{2}$ improved to 1 (by completely different methods).
