Minimal and maximal constituents of twisted Foulkes characters

We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms $s_\nu \circ s_{(m)}$. As a corollary we prove

two conjectures of Agaoka on the lexicographically least constituents of the plethysms $s_\nu \circ s_{(m)}$ and $s_\nu \circ s_{(1^m)}$.