  In the paper, we construct a representation $\theta:FVB_n\to{\rm
Aut}(F_{2n})$ of the flat virtual braid group $FVB_n$ on $n$ strands by
automorphisms of the free group $F_{2n}$ with $2n$ generators which does not
preserve the forbidden relations in the flat virtual braid group. This
representation gives a positive answer to the problem formulated by V. Bardakov
in the list of unsolved problems in virtual knot theory and combinatorial knot
theory by R. Fenn, D. Ilyutko, L. Kauffman and V. Manturov. Using this
representation we construct a new group invariant for flat welded links.
  Also we find the set of normal generators of the groups $VP_n\cap H_n$ in
$VB_n$, $FVP_n\cap FH_n$ in $FVB_n$, $GVP_n\cap GH_n$ in $GVB_n$, which play an
important role in the study of the kernel of the representation $\theta$.

Group Theory;Algebraic Topology