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Specified the chapter of the reference, since numbering of items in the cited book starts anew in each chapter.
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Stefan Kohl
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"Let $w_1$ and $w_2$ be elements of a free group $F$. Then it is decidable whether there is an automorphism of $F$ carrying $w_1$ into $w_2$."

(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Chapter I, Prop.4.19)

Is this an answer on your question? I think it's hard to get something more specific.

"Let $w_1$ and $w_2$ be elements of a free group $F$. Then it is decidable whether there is an automorphism of $F$ carrying $w_1$ into $w_2$."

(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Prop.4.19)

Is this an answer on your question? I think it's hard to get something more specific.

"Let $w_1$ and $w_2$ be elements of a free group $F$. Then it is decidable whether there is an automorphism of $F$ carrying $w_1$ into $w_2$."

(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Chapter I, Prop.4.19)

Is this an answer on your question? I think it's hard to get something more specific.

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Boris Novikov
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"Let $w_1$ and $w_2$ be elements of a free group $F$. Then it is decidable whether there is an automorphism of $F$ carrying $w_1$ into $w_2$."

(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Prop.4.19)

Is this an answer on your question? I think it's hard to get something more specific.