Proof of equality of OLS projection matrix and GLS projection matrix

I’m struggling with the proof of the following proposition:

Given a $ n\times n$ symmetric, positive-semidefinite matrix $ \Omega$ , a $ n\times k$ matrix $ X$ such that $ rank(X)=k$ , and a matrix $ Q$ such that: $ $ \Omega X=XQ$ $

Prove the following: $ {{(X'{{\Omega }^{-1}}X)}^{-1}}X'{{\Omega }^{-1}}={{(X’X)}^{-1}}X’$

Here’s my work before getting stuck:

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I also tried playing with the Cholesky factorization of $ \Omega$ but always ended up getting stuck with an irreducible expression. What am I missing? (for context, this problem is from a graduate-level econometrics course)