Functor of continuation in Hilbert cube and Hilbert space

A $Z$-set in a metric space $X$ is a closed subset $K$ of $X$ such that each map of the Hilbert cube $Q$ into $X$ can uniformly be approximated by maps of $Q$ into $X\setminus K$. The aim of the paper is to show that there exists a functor of extension of maps between $Z$-sets of $Q$ [or $l_2$] to maps acting on the whole space $Q$ [resp. $l_2$]. Special properties of the functor are proved.