$.  Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{IA}(G)=C^*$ if and only if either $G'=Z(G)$ or $G'<Z(G)$, $d(G')=d(Z(G))$, $d(G/Z(G))=d(G/G')$ and $\exp(G')=var(G/Z(G),G/G')=\exp(G/Z(G))= var(G',Z(G))$