  In [3] is was shown that for any group $G$ whose rank (i.e., minimal number
of generators) is at most 3, and any finite index subgroup $H\leq G$ with index
$[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which
generates $G$. In this paper we extend this result to groups of rank at most 4.
We also extend this to groups $G$ of arbitrary (finite) rank $r$ provided all
the non-trivial divisors of $[G:core_G(H)]$ are at least $2r-1$. Finally, we
extend this to groups $G$ of arbitrary (finite) rank provided $H$ is malnormal
in $G$.
