What is the inverse of an inner automorphism?

In the first method, you are doing $(f_a(x))^{-1}$; that is, you are inverting the output of a homomorphism. By homomorphism properties, this is also the same as $f_a(x^{-1})$.

In the second method, you are actually solving for the inverse of the homomorphism $f_a$ itself, and not taking the inverse of a single element in its output. So, the second method is correct.

How is it shown that quintic equations can be solved by radicals and ultraradicals?

A composite number containing exactly 1993 digits "1" and one digit "7" [closed]

Content $0$ implies boundary has content $0$

(New Solution needed) Explicit series expansion for inverse of $e^{-x}\left(\frac{x^2}2+x+1\right)$

calculate $\cos(\alpha+\beta)$ and $\sin(\alpha+\beta)$

Hard problem to show almost sure boundedness

$\exists t\in \mathbb{R}, t>0$ such that $v_t = x - ty \ge 0$

How to prove inequaltiy for real numbers [closed]

Proof that in an integral domain every prime element is irreducible [duplicate]

Proving that $x-x^2$ is a contraction without mean value theorem

Generator/generators for cyclic group $\mathbb Z[i]/(11-8i)$, given the following isomorphism: [duplicate]

Decomposing a $\mathbb Z[i]$-module using Smith normal form