  Let G be a finite group. Let pi be a permutation from S{n}. We study the
distribution of probabilities of equality a{1} a{2}
...a{n-1}a{n}=a{pi{1}}^{epsilon{1}}
a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}},
when pi varies over all the permutations in S{n}, and epsilon{i} varies over
the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting
Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led
to a close connection to Hultman numbers. In this paper we generalize the
results, permitting epsilon{i} to be -1. We describe the spectrum of the
probabilities of signed permutation equalities in a finite group G. This
spectrum turns out to be closely related to the partition of 2^{n}*n! into a
sum of the corresponding signed Hultman numbers.
