Let G be a locally compact group, H a closed subgroup and 1\(\leq p\leq \infty\). Suppose that \(\pi\) is a strongly continuous isometric representation of H in a Banach space. After recalling the concept of the p-induced representation \(ind^ G\!_ H(p,\pi)\), the \(L^ p\)-version of unitarily induced representation in Hilbert spaces, the author gives a short proof of the theorem on induction in stages, which has previously been shown by R. A. Fontenot and I. Schochetman [Rocky Mt. J. Math. 7, 53-82 (1977; Zbl 0377.47019)] and H. Kraljevič [Glas. Mat., III. Ser. 4(24), 183-196 (1969; Zbl 0226.43010)]. Next, intertwining operators for two representations \(\pi_ 1\) and \(\pi_ 2\) of H and for the corresponding p-induced representations are studied. Clearly, every intertwining operator T for \(\pi_ 1\) and \(\pi_ 2\) defines a p-induced intertwining operator \(ind^ G\!_ H(p,T)\) for \(ind^ G\!_ H(p,\pi_ 1)\) and \(ind^ G\!_ H(p,\pi_ 2)\). It turns out that conversely a given intertwining operator S for the induced representations arises in this way if and only if S commutes with pointwise multiplication with \(L^{\infty}(G/H)\)-functions. The author presents examples and some applications of the p-inducing process and its continuity with respect to a certain topology. Among them are the so- called Herz majorization principles controlling the coefficients of p- induced representations and the p-induced principle series of connected semisimple Lie groups of rank 1, in particular SL(2,\({\mathbb{R}})\).