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In this paper we describe an infinite family of Cameron-Liebler line classesof ${\rm PG}(3,q)$ with parameter $(q^2 + 1)/2$, $q\equiv 1\pmod{4}$. Theexample obtained admits ${\rm PGL}(2,q)$ as an automorphism group and it isshown to be isomorphic to none of the infinite families known so far whenever$q \ge 9$.

Cameron-Liebler line classes of ${\rm PG}(3,q)$ admitting ${\rm PGL}(2,q)$