Let $\frak F_{\lambda}^2$ be the space of tensor densities of degree $\lambda\in \mathbb{C}$ on the supercircle $S^{1|2}$. We consider the space $\mathfrak{D}_{\lambda,\mu}^{2,k}$ of $k$-th order linear differential operators from $\frak F_{\lambda}^2$ to $\frak F_{\mu}^2$ as a module over the superalgebra $\mathcal{K}(2)$ of contact vector fields on $S^{1|2}$ and we compute the superalgebra $\mathcal{K}_{\lambda,\mu}^{2,k}$ of endomorphisms on $\mathfrak{D}_{\lambda,\mu}^{2,k}$ commuting with the $\mathcal{K}(2)$-action. We prove that this algebra is trivial except for $\lambda= 0$.