= Solution
\b[The fundamental case.] Set $\langle\phi\rangle=v e_4$, where $v\ne0$. Fixing this <vector> forces the fourth column of a <unitary matrix> to be $e_4$, and hence also fixes its fourth row. The remaining three-by-three block has <determinant> one. Therefore
$$
\boxed{\operatorname{Stab}(ve_4)=\{\operatorname{diag}(W,1):W\in SU(3)\}=H_3.}
$$
Every nonzero complex <vector> can be brought to this form by an <SU(4)> transformation: complete its normalized direction to a unitary <basis> and adjust the phase of a complementary <basis vector> to obtain <determinant> one. Thus a generic nonzero fundamental <vacuum expectation value> has a conjugate of $H_3$ as its exact unbroken <subgroup>. The particular embedded $H_3$ of the question is fixed only when the <vacuum expectation value> lies along $e_4$.
\b[The antisymmetric case.] With $\epsilon=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$, take
$$
\langle A\rangle=\begin{pmatrix}a\epsilon&0\\0&b\epsilon\end{pmatrix}.
$$
Since $V\epsilon V^T=\epsilon$ for $V\in SU(2)$, this <antisymmetric tensor> is fixed by $H_2$ for every $a,b$. Its transformation is $A\mapsto UAU^T$, so any stabilizing $U$ also preserves
$$
AA^\dagger=\operatorname{diag}(|a|^2I_2,|b|^2I_2).
$$
If $|a|\ne|b|$, its two <eigenspaces> force $U=\operatorname{diag}(V,\widetilde V)$ with unitary two-by-two blocks. For each nonzero block, $V\epsilon V^T=(\det V)\epsilon$ forces the block <determinant> to be one. If one block is zero, the <SU(4)> <determinant> condition forces the other block <determinant> to be one as well. Hence
$$
\boxed{|a|\ne|b|\quad\Longrightarrow\quad\operatorname{Stab}(\langle A\rangle)=H_2.}
$$
The <unitary skew-diagonalization of an antisymmetric matrix> shows that a generic complex $\mathbf6$ has two distinct paired <singular values>; choosing an <SU(4)> rather than a $U(4)$ <basis> merely leaves an allowed overall phase in $a,b$. Thus its generic <stabilizer> is conjugate to $H_2$.
If instead $|a|=|b|\ne0$, the form is proportional, up to unitary coordinates and phases, to a nondegenerate standard symplectic form. Its <stabilizer> is the <compact symplectic group> $USp(4)$, of <dimension> ten, containing $H_2$ properly. The explicit choice $a=b=1$ is a concrete counterexample to inferring an exact $H_2$ <stabilizer> from the existence of two singlets alone. For a real $\mathbf6$, the <real structure of the SU4 exterior square> imposes $b=\bar a$ in this form, so these equal magnitudes are compulsory. Every nonzero real $\mathbf6$ therefore has the larger <compact symplectic group> <stabilizer>. The zero <vacuum expectation value> preserves all of $SU(4)$.
\b[The adjoint cases.] The $H_2$-fixed and $H_3$-fixed traceless <matrices> have respectively the forms
$$
X_2=a\operatorname{diag}(1,1,-1,-1),\qquad
X_3=a\operatorname{diag}(1,1,1,-3),\qquad a\ne0.
$$
For a Hermitian adjoint <scalar field>, choose $a$ real. More generally a complex $a$ gives the same <stabilizers>. The repeated <eigenvalues> imply
$$
\boxed{\operatorname{Stab}(X_2)=S(U(2)\times U(2)),\qquad
\operatorname{Stab}(X_3)=S(U(3)\times U(1)).}
$$
Here $S$ means that the product of the block <determinants> is one. These <groups> contain respectively $H_2$ and $H_3$, together with an additional continuous $U(1)$. Explicit extra transformations are $\operatorname{diag}(e^{it}I_2,e^{-it}I_2)$ and $\operatorname{diag}(e^{it}I_3,e^{-3it})$. They commute with $X_2$ and $X_3$ for every $t$, proving that neither exact endpoint $H_2$ or $H_3$ occurs for a single adjoint <vacuum expectation value>.
Nor do these larger <subgroups> survive for a generic Hermitian traceless <matrix>: four distinct <eigenvalues> leave only the <maximal torus> $U(1)^3$. For a generic complex adjoint <matrix>, there need not even be a continuous <stabilizer>; generically its unitary <centralizer> is just the <centre of a group> $\mathbb Z_4$. Thus the displayed degenerate adjoint directions are special. All these statements concern the <stabilizer> of a possible <vacuum expectation value>; no unspecified <scalar potential> can be assumed to select a particular direction or a particular reality condition.
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