Let $G$ be a group, $H$ a subgroup, and $G/H$ the set of left cosets of $H$ in $G$. We can give this set an operation $*$, defined by $(g_1H) * (g_2H) = (g_1g_2)H$, which is well-defined if and only if $H$ is normal in $G$. In this case $(G/H, *)$ is a group.
However, even if $H$ is not normal, $G/H$ can still be given a group structure since every non-empty set admits a group structure (assuming the Axiom of Choice). I assume that there is no "natural" way to assign a group structure to $G/H$ that ends up being useful. In what sense, if any at all, can this last sentence be made precise? Can it be made more general (e.g. categorical)?