specific heat capacity

Now consider two objects,A and B. A has a higher specific heat capacity that B. When both object is subjected to same amount of thermal or heat energy, rise in temperature in A is lower as our common reason(A has a higher specific heat capacity)

But what stated by First law of thermodynamics that Increase in internal energy is caused by heat energy transfer to in and work done on it. since temperature is a measure of average kinetic energy of particles in a system, it means perticles in A has a greater amount of kinetic energy than B. Meanwhile , energy should be conserved, so it means portion
heat energy flow to A and is stored as kinetic energy of particles is lower than B.
NOW, WHERE DOES THE REST PORTION OF THE THERMAL ENERGY FLOW TO A converted to? the electrical potential energy(another component of internal energy) of the particles in A ?
if so that means object with high specific heat capacity has a lower tendency to store the thermal energy received in the form kinetic energy of particles?

can somebody explain this to me? or correct me if i am wrong. Thanks a lot.