For any matrix A and an orthogonal matrix Q, I can prove in the standard way that $$\|QA\|_2 = \|A\|_2 $$ using $$\|QA\|_2^2 = (QAx)^T(QAx) = (Ax)^T(Ax) = \|A\|_2^2 $$
However, I am unable to cancel Q, when the transformation is AQ, i.e. $$\|AQ\|_2^2 = (AQx)^T(AQx) = x^TQ^TA^TAQx$$ After this point I am unable to prove the same that $$\|AQ\|_2 = \|A\|_2 $$
Recall that the 2-norm for matrices is defined as $$\|B\|_2 \;\; =\;\; \sup_{\|x\|=1} \|Bx\|.$$
But for any orthogonal matrix $Q$ we have that $\|Qx\| = \|x\|$. Thus in your computation we can write $$\|AQ\|_2 \;\; =\;\; \sup_{\|x\|=1} \|AQx\| \;\; =\;\; \sup_{\|Qx\|=1} \|AQx\| \;\; =\;\; \sup_{\|y\|=1} \|Ay\| \;\; =\;\; \|A\|_2.$$