We study a variety of questions related to the Scott modules S(G,Q) associated to a finite group G, where Q denotes a p-subgroup of G for a given prime p. The main concept we study is that of a p-extendible group, which we define to be a group in which the dimension of S(G,Q) is minimal for all p-subgroups Q of G. We study those Frobenius groups which are p-extendible and complete a classification of the local subgroups of the sporadic groups which are p-extendible. Furthermore, we study Scott modules associated to finite classical groups which admit (B,N)-pairs that are split at characteristic p. The thesis concludes with some considerations about the second relative syzygy with respect to a subgroup Q for a certain class of p-groups P.