Several test groups and one big control group: split the group or not?

Dear members

I'm preparing a test of proportions regarding a campaign redemption of several groups which have all common "demographics" but have been stimulated into different ways.

At the same time I have a big group of people with the same demographics but which have not been stimulated into any way.

My question is, does it make sense to use the whole control group several times as control for each different test group, or instead should I split the control group into different sub-groups and then use each one in a one-to-one way with each test group?

Furthermore, there is any side effect in using a member of the total control group more than once in a sub control group?

Sure. I have two test group which have received two different type of stimulus in order to stimulate theyr redemption (binary response, 1 redemption, 0 no redemption). Also, I have a third bigger group of people which have received no stimulus in the same period of time. I want test if the proportion of the redemptors is the two tested groups is significatively different from the redemption rate in the control group. The doubt I have is if It make sense to

a. test separately each test group vs the contorl group, and in doing so using the whole control group two times, or
b. if it make sense to split the control group into two different sub control groups, and use each of them separately with each tested group, or,
c. now that i'm thinking about it, if it make sense to test all the three groups together.