If I have calculated correctly (which is not certain), there is an example of what you are asking for $G = C_4 = \langle g \rangle$ (cyclic of order $4$).
The normalized $2$-cocycle $t$ with $t([g,g^2]) = t([g^2,g]) = t([g^2,g^2]) = t([g^3,g^3]) = (1,1)$ and $t([g^i,g^j] = (0,0)$ otherwise is not a $2$-coboundary, and so does not lie in the image of $Z^2(G,Z/2)$.