A representation of $\mathbb{Z}$ is just a finite dimensional vector space over $k$ + an endomorphism. This is the same as a representation of the one-dimensional Lie algebra $k$. When $k$ is algebraically closed, the Tannakian fundamental group is known, but it is complicated, namely, it is $Speck[t]×Speck[k]$. See question 21415 (answers of Milne/Ekedahl).
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