Let \(G\) be a group and \(S\) be its normal subsemigroup. There has been considerable interest in extension of complex multipliers on \(S\) to multipliers on \(G\). The paper under review generalises the earlier work of \textit{G.\,J.\thinspace Murphy} [Proc.\ Am.\ Math.\ Soc.\ 125, No.\,1, 121--127 (1997; Zbl 0860.47003)] on this topic to the case of a unitary-valued multiplier \(\sigma\). As one of their results, the authors explicitly build a projective unitary \(\Phi(\sigma)\)-representation of \(G\) which is a dilation of a projective isometric \(\sigma \)-representation of \(S\).