How to pay transaction fee for another addresss?

I have a wallet where there are ERC-20 token and 0 eth. I also have some eth in another wallet. I want to transfer ERC-20 token to exchange.

Can I use eth in the 2nd wallet to pay gas fee for the transaction of the first one? so I dont have to double pay the gas fee (transfer some eth to the first wallet and transfer ERC-20 token out). or it is impossible?