Forcible groups and Frattini covers

We say that a finite group $G$ forces a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$, and we say $H$ is forcible if some finite group $G$ forces $H$. Complementing a classical result of Thompson–Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.