Fusion invariant characters of <i>p</i> -groups

We consider complex characters of a p-group P, which are invariant under a fusion system F on P. Extending a theorem of Brcenas-Cantarero to nonsaturated fusion systems, we show that the number of indecomposable Finvariant characters of P is greater or equal than the number of F -conjugacy classes of P. We further prove that these two quantities coincide whenever F is realized by a p-solvable group.On the other hand, we observe that this is false for constrained fusion systems in general.Finally, we construct a saturated fusion system with an indecomposable F -invariant character, which is not a summand of the regular character of P.This disproves a recent conjecture of Cantarero-Combariza.