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Let 
$\alpha $ be a complex-valued 
$2$-cocycle of a finite group G with 
$\alpha $ chosen so that the 
$\alpha $-characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also 
$\alpha $-regular are characterised by the values that the irreducible 
$\alpha $-characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of 
$G;$ however, the initial choice of 
$\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.
