A Tower Controller in Charge reported the Ground Controller cleared an aircraft to taxi across a runway at the same time another aircraft was departing. The departing aircraft flew over the top of the taxiing aircraft.

Narrative

I was working the Controller in Charge (CIC) position. The local control position had control of Runway XX and Runway XY. Aircraft X landed Runway XX turned left on A2 taxiway. Local Controller told the aircraft to contract ground. Aircraft Y called ready for takeoff on Runway XY. The Local Controller cleared Aircraft Y for take-off they began their roll. The Ground Controller taxied Aircraft X to the ZZZ via Alpha-Lima-Cross Runway XY. The pilot read it back as they were starting to taxi onto A. Another pilot called the Ground Control position for a clearance. The Local Controller verbally told the ground Controller hold short of Runway XY; hold short of Runway XY; I have traffic; hold short of Runway XY. I as CIC stated the same thing. Hold your vision jet short of Runway XY he has someone departing. Hold short of Runway XY.The Ground Controller turned and looked at both of us as we were telling her to hold short and looked confused. The Ground Controller turned back to her flight progress strip and continued issuing the IFR clearance. Aircraft X was never given a hold short instruction. Aircraft Y was already wheels up on departure in a critical phase of flight. As Aircraft X crossed Runway XY on Taxiway A; the Aircraft Y flew right over the top of Aircraft X in the intersection of Runway XY and Taxiway A. The Ground Controller then looked at the local controller and said 'oh no...I'm sorry; oopsie...ugh I shoulda used my memory aid; oh no I'm sorry'. The pilot of Aircraft X called the Ground Controller on frequency after they crossed the runway and were over flown by the aircraft to verify they were given a cross runway instruction. The Ground Controller replied with Affirmative. Aircraft Y that departed never mentioned anything of the incident. After it happened we re-told the Ground Controller that we said to hold short multiple times. At the time of this incident there were a few airport vehicle's and mowers on the field working with Ground Control. There was also a coyote running across the airport on the active runways and taxiways while there was traffic in the area. Eyes were on the coyote to make sure it wasn't going to run out on runway with departing aircraft and helping the vehicles find its location to run off field. The operation was a little more complex than our normal operations at ZZZ. Mandatory memory aid while a runway is being used/hot. Displayed on podium.