Cohen-like poset for adding automorphisms of Fraisse limits

There exists a natural forcing notion which turns the set of integers into a Fraisse limit of a given Fraisse class. This long-known phenomenon provided a rough intuition that Fraisse limits, as "generic structures", have some connections with forcing. I investigate a version of this forcing, which together with "generic" structure, provides us a "generic" automorphism of this structure. Despite the forcing-theoretic language, I am not going to use anything beyond Rasiowa-Sikorski Lemma.