Check if $L\in CFG$ then $L'\in CFG$
$L'=\{w|ww^R\in L\}$
So, I show counterexample. Let $L=\{a^ib^ja^ib^l|i,j,l \ge 0\}\cdot \{b^za^xb^za^y|x,y,z\ge 0\} = \{a^ib^ja^ib^lb^za^xb^za^y|x,i,j,l,y,z\ge 0\}$
Then $L\in CFG$
What about word $ww^r\in L$ ? Such the word is form: $ \{a^ib^ja^ib^jb^ja^ib^ja^i|x,i,j,l,y,z\ge 0\}$
Then $L'=\{a^ib^ja^ib^j|i,j\ge 0\}\notin CFG $
Finally, it is not true that, if $L\in CFG$ then $L'\in CFG$
Ok ?