How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$?

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 Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds

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 $Spin(10)$?

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 Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds

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 $Spin(10)$?