Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ a subgroup of $G$ and s$\theta\in \hbox{\rm Irr}(H)$ an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $G$ induced by $\theta$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p\,{+}\,1}{2}$ distinct irreducible constituents.