Embeddings into highly transitive and mixed identity free groups

Given a countable group $G$, we develop a method to construct an overgroup $H$ that is finitely generated, highly transitive and mixed identity free. Our construction can be controlled to ensure that some fundamental group theoretic properties of $G$ are inherited by $H$, such as amenability or the property of not containing a nonabelian free group. The former provides a strong solution to a question of Hull and Osin, and the latter provides the first examples of nonamenable groups without free subgroups that are highly transitive and mixed identity free. Our examples also have a nontrivial amenable radical, answering a question of Arzhantseva.