The monomial Burnside biset functor on p-groups has composition factors whose minimal groups are cyclic p-groups or direct products of a cyclic p-group with a cyclic group of order p, with identified simple C[Aut(H)]-modules at those groups.
Bouc, Foncteurs d’ensembles munis d’une double action , Journal of Algebra 183 (1996), 664-736
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$p$-Biset Functor of Monomial Burnside Rings
The monomial Burnside biset functor on p-groups has composition factors whose minimal groups are cyclic p-groups or direct products of a cyclic p-group with a cyclic group of order p, with identified simple C[Aut(H)]-modules at those groups.