# Averaging an almost complex structure over compact Lie group action

It is well-known that if a Lie group $$G$$ acts on a symplectic manifold $$(W,\omega)$$ preserving $$\omega$$, then we can easily find a $$G$$-invariant almost complex structure $$J$$ by choosing first a $$G$$-invariant Riemannian metric -- that always exists by averaging an arbitrary one.

Then there is a unique $$A \in \Gamma(\mathrm{End}(TW))$$ that satisfies $$\langle v,w\rangle = \omega(v,Aw)$$ for all $$v,w \in TW$$. Both the metric and $$\omega$$ are $$G$$-invariant thus $$A$$ also must be, and then we basically only have to normalize $$A$$ to obtain the desired almost complex structure (in this case an $$\omega$$-compatible almost complex structure).

A similar method should work for taming almost complex structures since these are also in bijection to a convex domain of a vector space.

What about a general almost complex structures? If $$W$$ is a smooth manifold that admits almost complex structures, and if $$G$$ is a connected compact Lie group acting smoothly on $$W$$, is there any other method to obtain an almost complex structure on $$W$$ that is $$G$$-invariant?

Or are there any (easy) counter examples?