For a skew left brace ( G , ⋅ , ∘ ) , the map λ : ( G , ∘ ) → Aut ( G , ⋅ ) , a ↦ λ a , where λ a ( b ) = a − 1 ⋅ ( a ∘ b ) for all a , b ∈ G , is a group homomorphism . Then λ can also be viewed as a map from ( G , ⋅ ) to Aut ( G , ⋅ ) , which, in general, may not be a homomorphism. We study skew left braces ( G , ⋅ , ∘ ) for which λ : ( G , ⋅ ) → Aut ( G , ⋅ ) is a homomorphism. Such skew left braces will be called λ -homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism λ : ( G , ⋅ ) → Aut ( G , ⋅ ) gives rise to a skew left brace, which, indeed, is λ -homomorphic. As an application, we construct a lot of skew left braces (of infinite order) on free groups and free abelian groups . We prove that any λ -homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on λ -homomorphic skew left brace for which the image of λ is cyclic. We also obtain set-theoretic solutions of the Yang-Baxter equation corresponding to the skew braces we construct in this paper.