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Let $M$ be a smooth manifold on which $G$ acts smoothly, where $G$ is a compact Lie group. Suppose that $X=\bigcup_iS_i\subset M$ is a Whitney stratification with $G$-invariant strata, i.e. every stratum $S_i$ is a $G$-invariant submanifold of $M$. Then we have $$X/G=\bigcup_i(S_i/G)\subset M/G.$$ It is well known that, for every smooth $G$-manifold $M$, the orbit space $M/G$ is a Whitney stratifiction, so is $X/G$ a Whitney stratification as well ?

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