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) $? The kronecker product is where the confusion is setting in.
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This problem appears straightforward....?
From Steve's comment, There must be some typos in the statement of this problem. 

