Involutory decomposition of a group and twisted subsets with few involutions

A subset $K$ of some group $C$ is called twisted if $1\in K$ and $x,y\in K$ implies that $xy^{-1}x$ belongs to $K$. We use the concept of twisted subset to investigate and generalize the concept of involutory decomposition of a group. A group is said to admit involutory decomposition if it contains some involution such that the group is the product of the centralizer of the involution and the set of elements inverted by the involution. We study the twisted subsets with at most one involution. We prove that if a twisted subset has no involutions at all then it generates a subgroup of odd order.