  To $2$-categorify the theory of group representations, we introduce the
notions of the $3$-representation of a group in a strict $3$-category and the
strict $2$-categorical action of a group on a strict $2$-category. We also
$2$-categorify the concept of the trace by introducing the $2$-categorical
trace of a $1$-endomorphism in a strict $3$-category. For a $3$-representation
$\rho$ of a group $G$ and an element $f$ of $G$, the $2$-categorical trace
$\mathbb{T}r_2 \rho_f $ is a category. Moreover, the centralizer of $f$ in $G$
acts categorically on this $2$-categorical trace. We construct the induced
strict $2$-categorical action of a finite group, and show that the
$2$-categorical trace $ \mathbb{T}r_2$ takes an induced strict $2$-categorical
action into an induced categorical action of the initia groupoid. As a
corollary, we get the $3$-character formula of the induced strict
$2$-categorical action.
