GFCI not tripping with tester

On an inspection yesterday, the GFCI in the kitchen would not trip when I used my 3 light tester. When I tried the test button, it did trip and did not have power. When I plugged my tester into another receptacle, the GFCI did trip.

Why would it trip at another receptacle but not when I plug the tester directly into it? I have encountered similar before but in those cases, the GFCI still had power when it was tripped.