What do a control group and an experiment group have in common?

Whatever the focus of the experiment is, plus any others factors that might influence the outcome of the experiment. If you are testing a new cancer drug, the experimental group and the control group must both be people with the same type of cancer, and both be a representative distribution of the population, all races, genders, ages, etc. You want the only difference in the two groups to be what you are studying, i. e. the effects of the drug.

Whatever the focus of the experiment is, plus any others factors that might influence the outcome of the experiment.