Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with 


$$M\subseteq N \subseteq M[G]$$


and that $N=M(x)$ for some $x\in N$; i.e., it is the smallest inner model of $M[G]$ which contains $x$ and $M$.


Is $N$ a symmetric extension of $M$?



Models of ZF intermediate between a model of ZFC and a generic extension
Yes, if $N=M(x)$ (taking the modern notation over Grigorieff's $M[x]$), then it is a symmetric extension. This is a very recent result of Toshimichi Usuba (see [this](https://arxiv.org/abs/1904.00895) and [that](https://arxiv.org/abs/1912.10246)).


However, by the Bristol model construction, this symmetric extension need not be obtained by the forcing that was used to get $M[G]$. 


