  Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the
bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear
cohomological Mackey $2$-motives over a commutative ring $k$. They showed that
$P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives
by using the ring homomorphism from the crossed Burnside ring of a finite group
$G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]).
We study the behavior of motivic decomposition of cohomological Mackey
$2$-motives as images by $P$ of motivic decomposition of Mackey $2$-motives.
