I have a $T \times 3$ matrix $X$ where the columns are orthonormal. Then I have $N$ orthonormal rotation matrices $Q_i$ where $i = 1, ..., N$. All matrices $Q_i$ are orthonormal and of size $3 \times 3$. $Q$ is the (element-by-element) mean of all matrices $Q_i$.
Is the product $XQ$ also orthonormal?