Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant 2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in \{0,1,\ldots,n-1\}$, let $\Box ^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$.