Is there a nice character Theory for quaternionic representations of finite groups?

By this I mean a description of the number of Quaternionic representations of a finite group in terms of Conjugacy classes/ equivalence classes of elements in G of some sort, in analogy with the real character theory in terms of conjugacy classes of real elements?

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    $\begingroup$ One thought is that it might be possible to do something along these lines if you knew how many square roots (in $G$) each element of $G$ has. But in practice, this seems an unrealistic requirement. But, for the record, an irreducible character $\chi$ of $G$ has F-S indicator $-1$ if and only if $\sum_{g \in G}(\#$ square roots of $g$ )$\chi(g^{-1}) = -|G|$). $\endgroup$ – Geoff Robinson Jan 21 '17 at 5:51

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