The problem of characterizing groups that are union of conjugates of a proper subgroup was considered in some papers by Wiegold and others. In particular, you can look at

Transitive groups with fixed point free permutations, Archiv der Mathematik 27 (1976), 473-475.

Groups covered by conjugated of proper subgroups, Journal of Algebra 293 (2005), 261–268.

It turns out that the answer to your question is yes.

In fact, already the free group on two generators $F_2 = \langle a, b \rangle$ can be covered by the conjugates of one of its proper subgroups. This is shown in the first paper I linked, Example 3.1 page 474.

The problem of characterizing groups that are union of conjugates of a proper subgroup was considered in some papers by Wiegold and others. In particular, you can look at

Transitive groups with fixed point free permutations, Archiv der Mathematik 27 (1976), 473-475.

Groups covered by conjugated of proper subgroups, Journal of Algebra 293 (2005), 261–268.

It turns out that the answer to your question is yes.

In fact, already the free group on two generators $F_2 = \langle a, b \rangle$ can be covered by the conjugates of one of its proper subgroups. This is shown in the first paper linked above, see Example 3.1 page 474.

This problem was considered by some papers by Wiegold and others. In particular, you can look at

Transitive groups with fixed point free permutations, Archiv der Mathematik 27 (1976), 473-475.

Groups covered by conjugated of proper subgroups, Journal of Algebra 293 (2005), 261–268.

It turns out that the answer to your question is yes. In fact, already the free group on two generators $F_2 = \langle a, b \rangle$ can be covered by the conjugates of one of its proper subgroups. This is shown in the first paper I linked, Example 3.1 page 474.

The problem of characterizing groups that are union of conjugates of a proper subgroup was considered in some papers by Wiegold and others. In particular, you can look at

Transitive groups with fixed point free permutations, Archiv der Mathematik 27 (1976), 473-475.

Groups covered by conjugated of proper subgroups, Journal of Algebra 293 (2005), 261–268.

It turns out that the answer to your question is yes. 

In fact, already the free group on two generators $F_2 = \langle a, b \rangle$ can be covered by the conjugates of one of its proper subgroups. This is shown in the first paper I linked, Example 3.1 page 474.