On the vertices of indecomposable modules over dihedral 2-groups

Let k be an algebraically closed field of characteristic 2. We calculate the vertices of all indecomposable kD8-modules for the dihedral group D8 of order 8. We also give a conjectural formula of the induced module of a string module from kT0 to kG where G is a dihedral group of order ⩾8 and where T0 is a dihedral subgroup of index 2 of G. Some cases where we verified this formula are given.