Reading M.Isaacs book Finite Group Theory (FGT) I found the following statement.
"If a group $G$ has a normal Sylow $2$-subgroup, it is easy to see that every subgroup of $G$ has a normal Sylow $2$-subgroup".
My question. How one can sufficiently easy see this?
I try to be more concrete. Is it really important that we deal with $2$-groups, or it doesn't matter?
I can prove this for arbitrary $p$ (not necessary $2$) using the following fact (excercise 2.B.6) from FGT. I give it here.
A group $G$ has a normal Sylow $p$-subgroup iff every subgroup of the form $<x,y>$ has a normal Sylow $p$-subgroup, where $x,y$ are conjugate elements of $G$ having $p$-power order.
But to to prove the last statement I have used Baer's theorem, which is not very easy itself. So it seems to me that the author meant that there is more simple proof of the fact about normal Sylow $2$-subgroups.