Extending maps by injective $\sigma$-$Z$-maps in Hilbert manifolds

The aim of the paper is to prove that if $M$ is a metrizable manifold modelled on a Hilbert space of dimension $\alpha \ge {\mathrm{\aleph }}_0$ and $F$ is its $\sigma$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than $\alpha$ and its closed subset $A$, for any map $f:X\to M$, each open cover $\mathcal{U}$ of $M$ and a sequnce $(A_n{)}_n$ of closed subsets of $X$ disjoint from $A$ there is a map $g:X\to M$ $\mathcal{U}$-homotopic to $f$ such that $g{\mid }_A=f{\mid }_A$, $g{\mid }_{A_n}$ is a closed embedding for each $n$ and $g(X\setminus A)$ is a $\sigma$-$Z$-set in $M$ disjoint from $F$. It is shown that if $f(\mathrm{\partial }A)$ is contained in a locally closed $\sigma$-$Z$-set in $M$ or $f(X \setminus A) \cap \bar{f(\partial A)} = \empty$,themap$g$maybetakensothat$g\bigr|_{X \setminus A}$ be an embedding. If, in addition, $X\setminus A$ is a connected manifold modelled on the same Hilbert space as $M$ and $\overline{f(\mathrm{\partial }A)}$ is a $Z$-set in $M$, then there is a $\mathcal{U}$-homotopic to $f$ map $h:X\to M$ such that $h{\mid }_A=f{\mid }_A$ and $h{\mid }_{X\setminus A}$ is an open embedding.