Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*}C$, where $C$ is a constant sheaf on a geometric (or Nisnevich) not necessarily closed (!!) point $x$ of $X$, $i_x:x\to X$ is the corresponding morphism. In this particular case the statement seems easy to prove; yet I wonder whether it follows from some general statement, and what are the 'standard' references for this. Are any additional restrictions needed here?

Also, I wonder whether sheaves of the type $i^*i_{x*}$ were studied somewhere in the literature?