There is a standard additive decomposition of the Hochschild cohomology ring of the group algebra of a finite group $G$ as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of $G$. A special case of our main result describes the cup product in terms of this decomposition. As applications, we determine presentations for the Hochschild cohomology rings of

[(1)] the mod-$3$ group algebra of the symmetric group $S_3$,

[(2)] the mod-$2$ group algebra of the alternating group $A_4$, and

[(3)] the mod-$2$ group algebras of the dihedral $2$-groups.