Toeplitz–Schur multipliers of $S_p(L^2(G))$ for $p<1$

We study Toeplitz–Schur multipliers of Schatten–von Neumann class $S_p$ for $0<p<1$. We describe all functions $F$ on an arbitrary commutative locally compact group $G$ satisfying the following condition: for any integral operator in $S_p$ with kernel function $k(x,y)$, the kernel function $F(x-y)k(x)k(y)$ determines also an integral operator in $S_p$.