  We compute the mapping class group orbits in the homotopy set of framings of
a compact connected oriented surface with non-empty boundary. In the case $g >
1$ the computation is some modification of Johnson's results and certain
arguments on the Arf invariant, while we need an extra invariant for the genus
$1$ case. In addition, we discuss how this invariant behaves in the relative
case, which Randal-Williams studied for $g > 1$.
