  In this paper we describe the structure of a group of conjugating
automorphisms $C_n$ of free group and prove that this structure is similar to
the structure of a braid group $B_n$ with $n>1$ strings. We find the linear
representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of
2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the
braid group $B_3(P^2)$ of the projective plane are linear. Using result of J.
Dyer, E. Formanek, E. Grossman and the faithful linear representation of
Lawrence--Krammer of $B_4$ we construct faithful linear representation of the
automorphism group $Aut(F_2)$.
