What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product and the $diag$ is where the confusion is setting in.
Hints:
i) $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]= \left[ I \otimes \mathbb{E} \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$
ii) I guess $ZZ^T \sim nWishart(.)$, where $n$ is the number of rows in $Z$, but am not very sure.
I do not see what the Kronecker product is doing there, it is just creating a block-diagonal matrix were each diagonal block is a copy of the second factor.
The component of $diag(ZZ^T\mathbf1)-ZZ^T$ at position $(i,j)$ is $$ \left(\sum_kZ_k\right) Z_i\delta_{ij}-Z_iZ_j $$ It should now be trivial to compute the expectation and come up with zero.