Osorno formula for Ganter–Kapranov 2-characters of crossed modules

Let $\mathcal{K}=(H\xrightarrow{\partial}G)$ be a crossed module whose fundamental group $\pi_1(\mathcal{K})$ is finite. Let $\Theta$ be a 2-representation of $\widetilde{\mathcal{K}}$, and let $\mathfrak{X}_\Theta({\mathbf{b}},{\mathbf{a}},h)$ denote its Ganter–Kapranov 2-character. Osorno formula conjecture. There exists an Osorno formula for the value

$$\mathfrak{X}_\Theta({\mathbf{b}},{\mathbf{a}},h).$$

Such a formula would extend the corresponding formula for finite groups to crossed modules and facilitate a cohomological description of 2-representations and their 2-characters.