Harborth constants for certain classes of metacyclic groups

The Harborth constant of a finite group $G$ is the smallest integer $k\ge exp\left(G\right)$ such that any subset of $G$ of size $k$ contains $exp\left(G\right)$ distinct elements whose product is 1. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n,m}=〈x,y\mid x^n=1,y^2=x^m,yx=x^{?1}y〉$. We also solve the “inverse” problem of characterizing all smaller subsets that do not contain $exp\left(H_{n,m}\right)$ distinct elements whose product is 1.