Well...think about it. If she wants (accepts) $50/week and you give it to her, then your husband is guaranteed a ride and what she does with the money is up to her.
For someone you don't trust that much--are you planning on GIVING her the gas credit card (sounds like it if you want to make sure she can't use it on anything else) or using it yourself (bad idea, she'll see that as a cash cow!).
How about this idea? You drive your husband the 5 miles to your mom's place in the morning.
OR, like another poster said--gas up the Explorer yourself (or have your husband do it).
Sounds like there's a lot more going on here....$50/week for $50 miles seems a bit high.
*ETA* after your So What Happened. I still think you'd be nuts to give a gas credit card to someone who has proven that she is irresponsible with financial planning. Can you get her $50 gas gift cards each week?