Its (may be) asking about the generalization of this question.
Let $f : (\Bbb F,+) \to (\Bbb F,+)$ be a non-zero homomorphism. Pick out the true statements:
a. $f$ is always one-one.
b. $f$ is always onto.
c. $f$ is always a bijection.
d. $f$ need be neither one-one nor onto.
That is, what can you say about if the $\mathbb Q$ is replaced by a Field $\mathbb F$? We can discuss the both finite and infinite field? What about the continuity?
In general, any field is a vector space over $\mathbb F_p$ or $\mathbb Q$. So you can write the basis of $\mathbb F$ over the appropriate field and send the elements of that basis to any values in $\mathbb F$ with at least one value is non-zero.
(a,b,c) Are all only true if $\mathbb F\cong\mathbb F_p,$ for some prime $p$, or $\mathbb F\cong\mathbb Q$.
(d) Is true for all other fields.
Not sure what is meant about continuity, but most homomorphisms $(\mathbb R,+)\to(\mathbb R,+)$ are not continuous, and same for $\mathbb C$.