For each positive integer \(k\), let us consider the evaluation map \(\Lambda X \to X^k\) which sends a loop to its values at the \(k\)-th roots of unity \(\in S^1=\{\alpha\in \mathbb C:|\alpha |=1\}\), where \(\Lambda X\) denotes the free loop space on \(X\). Since this map is equivariant with respect to the action of the cyclic group \(C_k\) of \(k\) elements, the author hopes to study the induced map in \(C_k\)-equivariant cohomology. In particular, in this paper, for \(k=2^m\) with \(m\geq 1\), he computes the map induced by evaluation in \(C_k\)-equivariant cohomology with \(\mathbb Z/2\)-coefficients in terms of the approximation functor \(l\) [\textit{M. Bökstedt} and the author, Fundam. Math. 162, No. 3, 251-275 (1999; Zbl 0952.55006)].