${\rm PL}$ involutions on the nonorientable $2$-sphere bundle over $S\sp{1}$

We show that there are exactly nine distinct ${\text{PL}}$ involutions on the nonorientable $2$-sphere bundle over ${S^1}$, up to ${\text{PL}}$ equivalences. This, together with results of [1], [3] and [8], classifies all ${\text{PL}}$ involutions on the $2$-sphere bundles over ${S^1}$.