  For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient
link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L)
\otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta, \theta^{-1}] \rightrightarrows
E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta,
\theta^{-1}].$ This spectral sequence splits along quantum and $sl_2$ weight
space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank\
AKh^{2j-k,k} (\tilde L)$ for every pair of quantum and $sl_2$ weight space
gradings $(j,k)$. We also present a few decategorified consequences and discuss
partial results toward a similar statement for the Khovanov homology of
2-periodic links.
