Wikipedia provides a valuable summary of necessary and sufficient conditions and their use in natural language. Here is an overview:
In logic, necessity and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because P cannot be true unless Q is true. Similarly, "P is sufficient for Q" because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.
That article also provides a diagram mapping the relationship. One can simplify that diagram as follows:
If F is a sufficient condition for G, then the region associated with F is completely contained within the region associated with G as in the diagram. Note there may be other conditions, that do not include F or overlap with F, that are also sufficient for G. The diagram also shows that G is a necessary condition for F. Any other necessary conditions for F would have to overlap with G.
Similarly, the diagram shows the relationship between ~G and ~F. Note that the region associated with ~G is contained within the region associated with ~F. So ~G is a sufficient condition for ~F.
Now consider the statement in question:
"If having feature F is a sufficient condition for having feature G, then lacking F is not a sufficient condition for G."
If lacking F, or ~F, is a sufficient condition for G, then the region associated with ~F in the diagram would have to lie entirely within the region associated with G. That might be the case if G contained everything, that is, if G were a necessary condition for both F and ~F, a tautology, but in general it is not. We would also need to take precautions if G were a contradiction, always false. However, there is usually a region outside G, or ~G. Therefore, ~F, or lacking F, is not a sufficient condition for G, as the statement claims.
Wikipedia contributors. (2019, October 8). Necessity and sufficiency. In Wikipedia, The Free Encyclopedia. Retrieved 16:08, October 9, 2019, from https://en.wikipedia.org/w/index.php?title=Necessity_and_sufficiency&oldid=920256263