Water ingress through Extractor fan vent pipe

It sounds like your cavity is breached and water is transferred from out wall to inner wall . Without looking i could say for sure if its the pipe it self or a breach higher up the wall . First i would check if the pipe has a fall . Falling from the inside wall and dropping lower on out side wall . If this is the case it's likely the problem is higher and above the pipe . If the fall is towards the internal then this is likely the problem .

Assuming it is the cowl, and it has been solvent welded (glued), you will have to cut the pipe just below the cowl to fit new mushroom type fitting. If you wish to check the cowl is the problem, an upturned bucket(or similar) placed over the cowl during the next storm should give you some indication.

Cut the cage off which is wrong Glue mushroom vent on which is right

I've come across this problem quite often and during heavy rain and high winds, rainwater is blown into the vent pipe. You can buy and fit a dome shaped mushroom vent cap, but the best solution is to fit a 90 degree bend to the pipe and then fit another 90 degree bend to the first bend to make a U shape pointing down. Glue a vent cap onto the end of the bend to keep birds out. To keep insects out, fit some fine wire mesh to the inside of the vent cap. To fit a new solvent weld fitting, cut the old one off flush with its socket.

You need a mushroom cowl to shed rainwater, the terminal you have is a soil pipe terminal