Suppose that a finite group \(G\) acts on two sets \(X\) and \(Y\), and that \(FX\) and \(FY\) are the natural permutation modules for a field \(F\). We examine conditions which imply that \(FX\) can be embedded in \(FY\), in other words that \((\ast)\): There is
an injective \(G\)-map \( FX \rightarrow FY\). For primitive groups we show that \((\ast)\)
holds if the stabilizer of a point in \(Y\) has a `maximally overlapping’ orbit on \(X\). For groups of rank three, we show that \((\ast)\) holds unless a specific divisibility condition on the eigenvalues of an orbital matrix of \(G\) is satisfied. Both results are obtained by constructing suitable incidence geometries.

A Latin square \((S, \ast)\) is said to be \((3,2,1)\)-conjugate-orthogonal if \(x \ast y = z \ast w\), \(x \ast_{321} y\), \(z \ast_{321} w\) imply \(x = z\) and \(y = w\), for all \(x, y, z, w \in S\), where \(x_3 \ast_{321} x_2 = x_1\) if and only if \(x_1 \ast x_2 = x_3\). Such a Latin square is said to be \emph{holey}(\((3,2,1)\)-HCOLS for short) if it has disjoint and spanning holes corresponding to missing sub-Latin squares.
Let \((3,2,1)\)-HCOLS\((h^n)\) denote a \((3,2,1)\)-HCOLS of order \(hn\) with \(n\) holes of equal size \(h\). We show that, for any \(h \geq 1\), a \((3,2,1)\)-HCOLS\((h^n)\) exists if and only if \(n \geq 4\), except \((n,h) = (6,1)\) and except possibly \((n,h) = (6,13)\). In addition, we show that a \((3,2,1)\)-HCOLS with \(n\) holes of size \(2\)
and one hole of size \(3\) exists if and only if \(n \geq 4\), except for \(n = 4\) and except possibly \(n = 17, 18, 19, 21, 22\) and \(23\). Let \((3,2,1)\)-{ICOILS}\((v, k)\) denote an idempotent \((3,2,1)\)-COLS of order \(v\) with a hole of size \(k\). We provide \(15\) new \((3,2,1)\)-ICOILS\((v, k)\), where \(k = 2, 3,\) or \(5\).

A balanced part ternary design (BPTD) is a balanced ternary design (BTD) with a specified number of blocks, say \(b_2\), each having repeated elements. We prove some necessary conditions on \(b_2\) and show the existence of some particular BPTDs. We also give constructions of infinite families of BPTDs with \(b_1 = 0\), including families of ternary \(t\)-designs with \(t \geq 3\).