The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^{1}(\mathbb{K}\left(1\right|3),\mathrm{\U0001d52c\U0001d530\U0001d52d}(2,3);{\mathcal{D}}_{\lambda ,\mu}\left({S}^{1|3}\right))$ of the contact Lie superalgebra $\mathbb{K}\left(1\right|3)$ with coefficients in the space of differential operators ${\mathcal{D}}_{\lambda ,\mu}\left({S}^{1|3}\right)$ acting on tensor densities on ${S}^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left({X}_{f}\right)={D}_{1}{D}_{2}{D}_{3}\left(f\right){\alpha}_{3}^{1/2}$, ${X}_{f}\in \mathbb{K}\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{\U0001d52c\U0001d530\U0001d52d}(2,3)$ of $\mathbb{K}\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group...