  In this paper, two-to-one mappings and involutions without any fixed point on
finite fields of even characteristic are investigated. First, we characterize a
closed relationship between them by implicit functions and develop an AGW-like
criterion for 2-to-1 mappings. Using this criterion, some new constructions of
2-to-1 mappings are proposed and eight classes of 2-to-1 mappings of the form
$(x^{2^k}+x+\delta)^{s}+cx$ are obtained. Finally, a number of classes of
involutions without any fixed point are derived from the known 2-to-1 mappings
by the relation between them.
