integer network flow with symmetry

Question

Suppose we have a weighted directed graph $G=(V,E,f)$. Each $e\in E$ is associated with $f_e\in \mathbb{N}$. There is a source node $s$, which only has outgoing edges, and a sink node $t$, which only has incoming edges.

Consider the automorphism group of $G$, denoted as $A$.

Is there always an integer flow $F$

1. is a max flow, i.e., the value of the flow equals the min-cut of $G$.

2. is invariant under the action of any element in $A$.

Answer 1

No. Consider the graph with vertices $\{s, a, a', b, t\}$ and edges $\{s\to a, s\to a', a\to b, a'\to b, b\to t\}$. If each edge has weight 1 then the max integer flow is not invariant under the only non-identity automorphism.