UPDATES TO George M. Bergman, Supports of derivations, free factorizations, and ranks of fixed subgroup in free groups, Trans. Amer. Math. Soc., 351 (1999) 1531-1550.
Warren Dicks has found that Example 16, which he and Stallings showed me, had already appeared in the literature:

  • R. G. Burns, T. C. Chau and D. Solitar, On the intersection of free factors of a free group, Proc. Amer. Math. Soc. 64 (1977) 43--44. MR 56#5717 .

    Question 19 is answered in the negative by part (ii) of Theorem A of

  • Ilir Snopce, Slobodan Tanushevski, and Pavel Zalesski, Retracts of free groups and a question of Bergman, 2019, 12 pp., arxiv.org/pdf/1902.02378.pdf .

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