Let $F=F(x_1,..x_n)$ be a free algebra of a homogeneous variety of linear algebras freely generated by $X={x_1,...x_n}$ , $End F$ be the semigroup of endomorphisms of $F$ and Aut End $F$ be the group of automorphisms of the semigroup $End F$. We investigate the structure of the group Aut End $F$ for the variety of Lie algebras, the variety of $m$-nilpotent associative algebras, the variety of commutative algebras over so-called $R_1MF-$domains. These domains contain, in particular, Bezout domain, unique factorization domains. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of nxn matrices over $R_1MF-$domains ! is obtained.