= Solution
Five examples are
$$
C_{12},\qquad C_6\times C_2,
\qquad A_4,\qquad D_{12},\qquad \operatorname{Dic}_3,
$$
where $D_{12}$ denotes the symmetry <group> of a hexagon and $\operatorname{Dic}_3$ is the $3$-dicyclic <group>.
The first two are abelian, while the last three are not. Of the first two, only $C_{12}$ is cyclic. Among the nonabelian <groups>, $A_4$ has no element of order $6$; both of the others do. The dihedral <group> $D_{12}$ has seven involutions, whereas $\operatorname{Dic}_3$ has the unique involution $a^3$ and also has elements outside $\langle a\rangle$ of order four. These invariants prove that all five <groups> are pairwise non-isomorphic.
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