Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = EndkG(kH ↑G), showing that it is a split extension of a nilpotent ideal by the group algebra kNG(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of kH ↑G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic.