  We study left invariant conformal Killing-Yano (CKY) $2$-forms on Lie groups
endowed with a left invariant metric. We classify all $5$-dimensional metric
Lie algebras carrying CKY tensors that are obtained as a one-dimensional
central extension of 4-dimensional metric Lie algebras endowed with a
invertible parallel skew-symmetric tensor. On the other hand, we also classify
$5$-dimensional metric Lie algebras with center of dimension greater than one
admitting strict CKY tensors. In addition, we determine all possible CKY
tensors on these metric Lie algebras. In particular, we exhibit the first
examples of CKY $2$-forms on metric Lie algebras which do not admit any
Sasakian structure.
