  This paper deals with the Weak Inverse Galois Problem which, for a given
field $k$, states that, for every finite group $G$, there exists a finite
separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to
explain how one can generically produce families of fields which fulfill this
problem, but which do not fulfill the usual Inverse Galois Problem. We show
that this holds for, e.g., the fields $\mathbb{Q}^{{{\rm{sol}}}}$,
$\mathbb{Q}^{{{\rm{tr}}}}$, $\mathbb{Q}^{{{\rm{pyth}}}}$, and for the maximal
pro-$p$-extensions of $\mathbb{Q}$. Moreover, we show that, for every finite
non-trivial group $G$, there exists many fields fulfilling the Weak Inverse
Galois Problem, but over which $G$ does not occur as a Galois group. As a
further application, we show that every field fulfills the regular version of
the Weak Inverse Galois Problem.
