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

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