Let $X$ be a $2n$-dimensional $C^{\infty}$-manifold, let $G$ be a finite group acting smoothly on $X$. Let $J: T_{\mathbb{C}}X\to T_{\mathbb{C}}X$ be a map such that $J^2=-id$ and $gJg^{-1}=J$ for any $g\in G$.

Let $T^{0,1}X$ be the $-1$ eigenspace of $J$. We call $J$ is integrable if $[T^{0,1}X,T^{0,1}X]\subset T^{0,1}X$.

My question is: do we have a $G$-equivariant Newlander-Nirenberg theorem? More precisely, if $J$ is integrable, then is it true that $X$ is a complex manifold with $G$-acting holomorphically?

Let $X$ be a $2n$-dimensional $C^{\infty}$-manifold, let $G$ be a finite group acting smoothly on $X$. Let $J: TX\to TX$ be a map such that $J^2=-id$ and $g_*Jg^{-1}_*=J$ for any $g\in G$.

We can extend $J$ to a $\mathbb{C}$-linear map $J: T_{\mathbb{C}}X\to T_{\mathbb{C}}X. $Let $T^{0,1}X$ be the $-1$ eigenspace of $J$. We call $J$ is integrable if $[T^{0,1}X,T^{0,1}X]\subset T^{0,1}X$.

My question is: do we have a $G$-equivariant Newlander-Nirenberg theorem? More precisely, if $J$ is integrable, then is it true that $X$ is a complex manifold with $G$-acting holomorphically?