Classes of binary matroids with small lists of excluded induced minors

In earlier work, we characterized the class of matroids with no $M(C_4)$ as an induced minor and the class of matroids with no member of $\{M(C_4),M(K_4)\}$ as an induced minor. In this paper, for every two matroids in $\{M(C_4),M(K_4\mathrm{\backslash }e),M(K_4),F_7\}$, we determine the class of matroids that have neither of the chosen pair as an induced minor. Additionally, we prove structural lemmas toward characterizing the class of matroids that do not contain $M(K_4)$ as an induced minor.