Find average sequence from a set of sequences
One way would be not to approach this as a calculation per session. Most data science solutions like to end up with a number, probability or classification. I suggest you structure your data differently so that you try to answer the question - what next action is likely given the last action.

In order to do this you would have to restructure your session data and use information from across all your sessions. For example, if you compare in how many sessions a player 'buys a gun', and if so record over all those sessions what their next action is, e.g. in 60% they 'play a mission' next. You will then have a probability of their next action based on the number of choices players made in all those sessions.

Once you have those probabilities, you will be able to answer the question, 'What comes next?'. This will in turn enable you to build that most average session that you are after by stepping through a session and building it by the most probable next step.