For every 4 ‘c’ in the input push a ‘c’ in the stack (Push down automata)
You can recognise the language $L = \{c^{4n}|n\geq 0\}$ with a normal DFA. Use this automata as a base and just use one designated state, that pushes a C onto your stack, for which I would choose the first one. For every transition one c is consumed from the word.

$\{(q_1, c, S, CS, q_2) ,
(q_2, c, S, S, q_3),
(q_3, c, S,S,q_4),
(q_4, c, S,S,q_1),
(q_1, d, C, \varepsilon, q_d)\} \subset \Delta $
with $S \in \Gamma$