See my replies in these threads first:
https://www.rollitup.org/t/seed-generations-f1-2-3-bx1-2-etc-info.841001/
https://www.rollitup.org/t/why-dont-seedbanks-state-the-if-the-seeds-are-f1-f2-f3-f4-ect.839749/page-2
Opposed to popular belief, maybe because of that crap cannabis breeding book, backcrossing doesn't automatically lead to stable for breeding. Backcrossing has very specific purposes, such as breeding "a" trait from an otherwise not so great pheno into a stable line, saving the
genes (but not the necessarily the
genotypes) from a clone only strain, or cubing / population backcrossing (which can lead to stability but only for the traits that are homozygous in the recurrent parent). Creating a stable homozygous line using backcrossing requires having already a stable homozygous IBL. It's like chicken and egg...
Homozygous stable lines are not the result of backcrossing, not the result of population breeding (mating the best with the best and cull the rest a la heirloom breeding) but the result of what actual breeders have been doing since a little over 100 years, pure line breeding. Cross two of those pure lines and you get a true F1 hybrid, where the F1 means more than just being the first offspring generation. Most of the popular veggies, fruits, and ornamental plants were created like that.
In the cannabis industry, it's like growing, people (think they can) make up their own ways and don't care about "modern plant breeding" based on Mendel's work.
The truth is, many F1s in the cannabis industry don't have the hybrid vigor / heterosis either because they are not the result of crossing two "homozygous" IBLs. They are F1s, but not really F1 hybrids.
The best approach depends all on your goals. Breeding starts with defining clear goals. If that is creating a new variety you can use an existing strain and backcross a trait in to that (make you own Kush/haze/skunk or w/e variety), or if you want to create that differs more than a few traits from the original parents follow these steps: generate variety (e.g. by creating an F2 or double cross from two different F1s), in that variety hunt for the best phenos, and then try to reproduce those phenos consistently (which is really what it is all about) by selecting and stabilizing "genotypes". Figuring out the genotypes opposed to just mating phenos is THE hard part especially with complete dominant traits (genotype AA and Aa are both the same pheno, while you want to cross AA with AA to get only AA and breed out 'a' and lock in AA.)
That last thing is what leads to homozygous stable 'traits', which has to be done for all relevant traits. At what generation all the relevant traits are stable depends on the selection and as I showed here can theoretically occur in F3 already:
https://www.rollitup.org/t/when-is-it-considered-a-new-strain.842812/#post-10907966 That would require few, preferably just one, traits of interest and a lot of plants. In practice it requires more generations. As I mentioned in another thread, you're not supposed to stabilize an F1 hybrid but its parents. A valid reason to take F1 to F2 is to generate variation from which to find something new to breed stable, or to breed something stable to create an F1 hybrid again. Breeding stable starts at the soonest at F3.
To answer your specific situation, I wouldn't inbreed/stabilize an F4 without "seeing" how stable it is. Cross it that test run already to see how traits inherit (dominant/recessive). The easiest way to test for stability is to self it. The result should have no pheno variation for it to be a true bred homozygous plant. In reality you will always see some variation, i.e. one may be a better yielder or a bit more frosty, but for the relevant traits (if breeding for taste that is taste and smell for example) there should be zero pheno variation. If there is, it means one or more genotypes in that F4 plant you selfed is heterozygous (for example Aa) and not homozygous (AA or aa). By observing the ratio of the variation you can gauge the genotypes and how they inherit. Needless to say, breeding with just a handful of plants is not possible. You can still make some fine beans of course, but a stable line...
And that's where backcrossing comes into play again. By backcrossing (multiple plants from the offspring) to a plant, that offspring will become more homozygous every back cross population for at least the traits that are homozygous in the recurrent plant, and can be done with relatively few plants. If for example you self that F4 and get very little variation, it would make a good candidate to backcross traits into.
Self-proclaimed breeders occasionally bring up Mendel and Punnet, but what they should really be looking into is
Wilhelm Johannsen's work, the Genotype-Phenotype Distinction, his work on pure-line breeding and the discovery of hybrid vigor...