Let $G$ be a topological group, and let $\text{TAut}(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the group of maps $f \colon G \to G$ that are simultaneously group automorphisms and self-homeomorphisms).

We wish to give $\text{TAut}(G)$ a reasonable topology in the following sense:

1. $\text{TAut}(G)$ becomes a topological group with respect to this topology.
2. This topology should interact with/depend on the topology of $G$ in some way, i.e. we can require that the natural action $\text{TAut}(G) \times G \to G$ is continuous.

In the case that $G$ is compact, it is known that giving $\text{TAut}(G)$ the compact-open topology satisfies the above conditions, where the compact-open topology has as a subbasis sets of the form $$V(C, U) = \{f \colon G \to G \mid f(C) \subseteq U\},$$ where $C, U \subseteq G$ are compact and open, respectively.

If $G$ is locally compact, we can instead give $\text{TAut}(G)$ the $g$-topology, which has as a subbasis sets of the form $$V(K, W) = \{f \colon G \to G \mid f(K) \subseteq W\},$$ where either $K$ or $G \setminus W$ is compact.

The cases where $G$ is compact or locally compact are discussed in this paper of Dijkstra and this paper of Arens. In fact, these two papers discuss the group of self-homeomorphisms $\text{Homeo}(X)$ for a space $X$ which is not necessarily a topological group, so the case for $G$ and $\text{TAut}(G)$ follows from that (since $\text{TAut}(G)$ is a subgroup of $\text{Homeo}(G)$.

My question is, for a general topological group $G,$ is there a good way to describe a topology on $\text{TAut}(G)$ satisfying the two conditions above? We can simply say: "give $\text{TAut}(G)$ the coarsest topology such that it becomes a topological group and the action $\text{TAut}(G) \times G \to G$ is continuous," but I am hoping for something more explicit than that.

$\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the group of maps $f \colon G \to G$ that are simultaneously group automorphisms and self-homeomorphisms).

We wish to give $\TAut(G)$ a reasonable topology in the following sense:

1. $\TAut(G)$ becomes a topological group with respect to this topology.
2. This topology should interact with/depend on the topology of $G$ in some way, i.e. we can require that the natural action $\TAut(G) \times G \to G$ is continuous.

In the case that $G$ is compact, it is known that giving $\TAut(G)$ the compact–open topology satisfies the above conditions, where the compact–open topology has as a subbasis sets of the form $$V(C, U) = \{f \colon G \to G \mid f(C) \subseteq U\},$$ where $C, U \subseteq G$ are compact and open, respectively.

If $G$ is locally compact, we can instead give $\TAut(G)$ the $g$-topology, which has as a subbasis sets of the form $$V(K, W) = \{f \colon G \to G \mid f(K) \subseteq W\},$$ where either $K$ or $G \setminus W$ is compact.

The cases where $G$ is compact or locally compact are discussed in Dijkstra - On Homeomorphism Groups and the Compact–Open Topology and Arens - Topologies for Homeomorphism Groups. In fact, these two papers discuss the group of self-homeomorphisms $\Homeo(X)$ for a space $X$ which is not necessarily a topological group, so the case for $G$ and $\TAut(G)$ follows from that (since $\TAut(G)$ is a subgroup of $\Homeo(G)$).

My question is, for a general topological group $G$, is there a good way to describe a topology on $\TAut(G)$ satisfying the two conditions above? We can simply say: "give $\TAut(G)$ the coarsest topology such that it becomes a topological group and the action $\TAut(G) \times G \to G$ is continuous," but I am hoping for something more explicit than that.