Should I be friendly and helpful to my girlfriend's best girl friend? Or could that feel like cheating?

I don't mean just saying hi and being friendly like with a stranger, but rather friendly like with my own best friend.

Not long ago, my girlfriend invited her BFF to hang out with us and to go to a show we had tickets to. But her girl friend doesn't live far away from me, so I offered to go pick her up before going to meet with my girlfriend, so that she wouldn't have to come alone at night with a bus or an Uber. I felt like if she is my girlfriend's friend, she is also my friend, so it felt right to offer to myself to drive her to the meet up. Of course, I didn't call her myself, I asked my girlfriend to ask her if she wouldn't want me to pick her up. But after this, my girlfriend said that she felt really off and jealous about letting me go pick up her friend, because she could have come by herself...

Another time, my girlfriend went out with her friend and it started raining, so she called me to come pick her up. We had to walk to the car, because I couldn't find any parking nearby, so I brought an extra umbrella for her friend and some gloves, since it was also very cold outside. She refused both, but the fact that I also thought about her, seemed to really bother my girlfriend. I also wanted to propose to drive her home, since it was heavily raining, but my girlfriend stopped me, saying that she can get home just fine by bus.

Is my girlfriend right that my friendliness acts seem too much like I am hitting on my girlfriend's best friend? Or is it normal to be this friendly, to ask her if I could be of any help?

by the way, my girlfriend's best friend also has a boyfriend, but he didn't want to come with us, nor did he offer himself to pick her up.

I was also thinking, for when we would meet again in the future, to also bring a chocolate, or something to eat, not only for my girlfriend, but also for her best friend (if she continues to come without her boyfriend). Would that be too much?

Should I be friendly and helpful to my girlfriend's best girl friend? Or could that feel like cheating?
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