Let \(U\) be a smooth connected affine curve over an algebraically closed field \(k\) of characteristic \(p>0\). An explicit description of the set of finite quotients of the étale fundamental group \(\pi_1(U)\) was conjectured in 1957 by Abhyankar, and proved by \textit{M. Raynaud} [Invent. Math. 116, 425--462 (1994; Zbl 0798.14013)] (in the affine line case) and by \textit{D. Harbater} [Invent. Math. 117, 1--25 (1994; Zbl 0805.14014)] (in general case), giving a necessary and sufficient condition for a finite group to be a Galois group of an étale cover of \(U\). This paper investigates questions of which covers of \(U\) are dominated by other covers having specified Galois groups, and of which inertia groups can arise over points ``at infinity''. The main results give constructions of modified covers of a given cover (enlarging a given Galois group by a quasi \(p\)-group, or enlarging the \(p\)-parts of inertia subgroups of a given Galois group) with special controls of branching data. As an application, a tame analogue of the geometric Shafarevich conjecture is proved. It implies, for example, the following result: Let \(\pi_1^t(U,\Sigma)\) be the Galois group of the maximal extension of the function field of \(U\) that is at most tamely ramified over the places in \(\Sigma\subset U\), and is étale over all places corresponding to other points of \(U\). If \(k\) is finite and \(\Sigma\) is a dense open subset of \(U\), then \(\pi_1^t(U,\Sigma)\) is isomorphic to the semidirect product of \(\hat\mathbb Z\) and the free profinite group of countably infinite rank.