Wang-Sun Formula in $\overrightarrow{GL}(\mathbb{Z}\mathrm{/}2k\mathbb{Z})$

Wang and Sun proved a certain summatory formula involving derangements and primitive roots of the unit. We study such a formula but for the particular case of the set of affine derangements in $\overrightarrow{GL}(\mathbb{Z}\mathrm{/}2k\mathbb{Z})$ and its subset of involutive affine derangements in particular; in this last case its value is relatively simple and it is related to even unitary divisors of $k$.