Vulnerabilities such as SQL injection represent a serious challenge to security. While tools with a pre-defined logic are commonly used in the field of penetration testing, the continually-evolving nature of the security challenge calls for models able to learn autonomously from experience. In this paper we build on previous results on the development of reinforcement learning models devised to exploit specific forms of SQL injection, and we design agents that are able to tackle a varied range of SQL injection vulnerabilities, virtually comprising all the archetypes normally considered by experts. We show that our agents, trained on a synthetic environment, perform a transfer of learning among the different SQL injections challenges; in particular, they learn to use their queries to efficiently gain knowledge about multiple vulnerabilities at once. We also introduce a novel and more versatile way to interpret server messages that reduces reliance on expert inputs. Our simulations show the feasibility of our approach which easily deals with a number of homogeneous challenges, as well as some of its limitations when presented with problems having higher degrees of uncertainty.