How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that *agree with a constraint on the mapping between their representations*?

By specifying the embedding, I mean that we can determine precisely the way how the Lie group embedding $G_1 \subset G_2$ is fixed as an embedding of two differentiable manifolds (since Lie groups are differentiable manifolds). Of course, as Lie group embedding, $G_1 \subset G_2$, the two groups must share the common identity element $\mathbf{1}$. So their group identity element
$
\mathbf{1}_{G_1} =\mathbf{1}_{G_2} 
$
are the same point on two manifolds.

Previously, I asked, is it enough to give some irreducible representation (irrep) of $G_1$ called $\mathbf{R}_{1,j}$ and some irrep of $G_2$ called $\mathbf{R}_2$, then we dictate the map
$$
\mathbf{R}_{1} =  \bigoplus_j \mathbf{R}_{1,j} \text{ in } G_1 \mapsto \mathbf{R}_2 \text{ in } G_2, \text{ and } G_1 \subset G_2 \tag{1}
$$
would the above be precisely enough to specify the embedding? Is this a necessary and sufficient condition? If not, what else data is needed? See https://mathoverflow.net/q/401944/27004


Here I would like to use a specific example to demonstrate whether we can uniquely specify the embedding or whether we can enumerate possible differnt embedding given the 
eq.(1).


0. Let us take a special unitary group $G_1=SU(5)$ into a Spin group $G_2=Spin(10)$. The is a lift map from $SU(5) \to SO(10)$ to $SU(5) \to Spin(10)$ which the universal cover $\pi_1(Spin(10))=0$ consistent with the lift map with $\pi_1(SU(5))=0$,
$$
\begin{array}{ccc}
  SU(5)  & \longrightarrow &  Spin(10)\\
          &\searrow &           \downarrow\\
  &  & SO(10).
\end{array}
$$


1. Let us specify a first possible way of embedding $SU(5)  \subset  Spin(10)$ via
$$
\mathbf 5 \oplus \overline{\mathbf{10}} \oplus  \mathbf 1  \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$
or
$$
\bigwedge{}^{1}\mathbb{C}^5 \oplus \bigwedge{}^{3}\mathbb{C}^5 \oplus  \bigwedge{}^{5}\mathbb{C}^5  \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).\tag{a}
$$
Here ${\mathbf 5}$ is the fundamental representation of $SU(5)$ in 
$\bigwedge{}^{1}\mathbb{C}^5$. 
Here  $\bigwedge$ is the wedge product of vectors in the vector space $\mathbb{C}^5$.

> **Question 1:** Does eq.(a) specify a unique embedding of $SU(5)  \subset  Spin(10)$? Or is it possible to have two or more such distinct $SU(5)$ embedding in $Spin(10)$ with the map given eq.(a)? If so, how are these $SU(5)$ different from each other?


2. Let us specify a second possible way of embedding $SU(5)  \subset  Spin(10)$ via
$$
\mathbf 1 \oplus  \mathbf{10}  \oplus \overline{\mathbf{5}}  \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$ 
or
$$
\bigwedge{}^{0}\mathbb{C}^5 \oplus \bigwedge{}^{2}\mathbb{C}^5 \oplus  \bigwedge{}^{4}\mathbb{C}^5  \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).
\tag{b}$$


> **Question 2:** Does eq.(b) specify a unique embedding of $SU(5)  \subset  Spin(10)$? Or is it possible to have two or more such distinct $SU(5)$ embedding in $Spin(10)$ with the map given eq.(b)? If so, how are these $SU(5)$ different from each other?
 
> **Question 3:** How are the embedding of eq.(a) and eq.(b) related to each other? I suppose, they are related by the outer automorphism of $SU(5)$ which is a $\mathbf{Z}/2$. 
Then, if so, how does this $\mathbf{Z}/2$ outer automorphism of $SU(5)$ act on the $Spin(10)$?