Hasse norm principle for metacyclic extensions with trivial Schur multiplier

Let $k$ be a global field, $K\mathrm{/}k$ be a finite separable field extension and $L\mathrm{/}k$ be the Galois closure of $K\mathrm{/}k$ with Galois groups $G=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}k)$ and $H=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K)⪇G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse norm principle holds for $K\mathrm{/}k$. We show that if $G$ is metacyclic with trivial Schur multiplier $M(G)=0$, then $H$ is cyclic and the Hasse norm principle holds for $K\mathrm{/}k$. Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quaternion, extraspecial groups and $Z$-groups $G$ with trivial Schur multiplier $M(G)=0$ are given. These provide new examples which the Hasse norm principle hold for non-Galois extensions $K\mathrm{/}k$ whose Galois closure is $L\mathrm{/}k$ with metacyclic $G=\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}k)$ and $M(G)=0$.