  We consider complex characters of a p-group P, which are invariant under a
fusion system F on P. Extending a theorem of B\'arcenas--Cantarero to
non-saturated fusion systems, we show that the number of indecomposable
F-invariant 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.
