Adapter socket works and doesn't works
It is possible that the outter shell (Screw portion) of the outside lamp is longer not allowing the adapter to screw all the way in and touch the center contact.
with the switch off(perferrably circuit breaker off) you might try pulling the center contact out so it will make contact. 
if this works you may need to place some insulating material behind the contact so it wont push back in when it gets warm from the light, 
Please if you do this make sure the circuit is turned off best checked with a volt meter or non contact "tick tester" so you dont light yourself up.