This article focuses on those aspects about partial actions of groups which are related to Schur's theory on projective representations. It provides an exhaustive description of the partial Schur multiplier, and this result is achieved by introducing the concept of a second partial cohomology group relative to an ideal, together with an appropriate analogue of a central extension. In addition, the new framework is proved to be consistent with the earlier notion of cohomology over partial modules.

For a finite group we introduce a particular central extension, the unitary cover, having minimal exponent among those satisfying the projective lifting property. We obtain new bounds for the exponent of the Schur multiplier relating to subnormal series, and we discover new families for which the bound is the exponent of the group. Finally, we show that unitary covers are controlled by the Zelmanov solution of the restricted Burnside problem for 2-generator groups.

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