Representation groups are a nice example. If $G$ is a finite group of order $n$ and if $m$ is the order of $H^2(G,\mathbb{C}^*)$, then a representation group is a group written as a central extension $$1\rightarrow A\rightarrow H\rightarrow G\rightarrow 1$$ such that $H$ has order $mn$ and every projective representation of $G$ lifts to a linear representation of $H$. It is a fact that representation groups always exist, although they are not unique. For instance, both the dihedral group $D_8$ of order $8$ and the quaternion group $Q_8$ are representation groups for $(\mathbb{Z}/2)^2$. I feel like these are particularly relevant because if $G$ is an abelian $p$-group with at least two cyclic factors, then any such $H$ must be non-abelian.