For language $L-$ $M=(Q,\Sigma,\delta, q_0, F) $
For language $L'-$ $M'=(Q',\Sigma,\delta, q_{start}, F')$
$Q'=(Q\times Q)\cup q_{start}$
Transistions
$\delta'(q_{start}, \epsilon)=\{(q,q_{acc})|q_{acc}\in F\}$
Transition from $(q_1, q_2)$ to $(q_3,q_4)$
$\delta'((q_1,q_2),a\in\Sigma) = (q_3,q_4)\text{ iff }\delta(q_1,a )=q_3 \text{ and } \delta(\delta(q_2,b),c)=q_4 $ for some $b,c\in\Sigma$
Accepting states $F'=\{(q,q)|q\in Q\}$

For language $L-$ $M=(Q,\Sigma,\delta, q_0, F) $
For language $L'-$ $M'=(Q',\Sigma,\delta, q_{start}, F')$
$Q'=(Q\times Q)\cup q_{start}$
Transistions
$\delta'(q_{start}, \epsilon)=\{(q,q_{acc})|q_{acc}\in F\}$
Transition from $(q_1, q_2)$ to $(q_3,q_4)$
$\delta'((q_1,q_2),a\in\Sigma) = \{(q_3,q_4)\}\text{ iff }\delta(q_1,a )=q_3 \text{ and } \delta(\delta(q_4,b),c)=q_2 $ for some $b,c\in\Sigma$
Accepting states $F'=\{(q,q)|q\in Q\}$