Let the statements given by $\text{A}$ be $\text{A1, A2}$ in order. Similarly, let us denote $\text{Bi, Ci, Di, i} \in \{1, 2\}$ for other statements. Since one of $\text{A, B, C, D}$ did the murder, three among $\text{A1, B1, C1, D1}$ are true. Observe that the statements $\text{A2, B2}$ are inverse to each other. Therefore one of them is true. It follows that the four true statements are from $\text{A1, B1, C1, D1, A2, B2}$. Hence, we could conclude that $\text{C2, D2}$ are false. This implies, $\text{D}$ did not commit the murder, and none of the men committed the murder. Further, out of the two women $\text{B, D};$ since $\text{D}$ has not committed the murder, we conclude that $\text{B}$ is the murderer.
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