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.