Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns, respectively. I would like to compute the following quantity: \begin{equation} \mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}], \end{equation} for $q\in\mathbb{N}$.

One can expand $a_{i}^{\top}AA^{\top}a_{j}$ to get \begin{equation} \|a_{i}\|_{2}^{2}a_{i}^{\top}a_{j}+\|a_{j}\|_{2}^{2}a_{i}^{\top}a_{j}+\sum_{t\in[k]\backslash\{i\}\cup\{j\}}(a_{i}^{\top}a_{t})(a_{j}^{\top}a_{t}) \end{equation} and explicitly compute the expectation for $q=1, 2$ etc. But how can I generalize for larger $q$?

[Posting this here since couldn't get answer on Math Stack Exchange and Cross Validated]