Short answer: unclear.
It strongly depends on where you are in the world, your vaccination status, sex/gender, age, co-morbidities and your access to medical facilities. If you take the case of the western world, the case fatality rate (CFR) for Omicron infection is between about 8.3 (table 3, 30 day CFR in 1) and 1.9 (2) per 100,000 cases; while the global CFR for Influenza is somewhere around 5.5 - 5.9 (3). From these data I would say on a par with Influenza.
However, it definitely isn't as simple as that, as you will see if you peruse reference 3. There are a huge number of factors that go into reporting these - for influenza there is a lot of modelling that goes on - the number of cases reported is nowhere near the actual numbers of cases because of widespread community transmission and a lack of routine community testing. Whereas for COVID, much of the data is as close to reality as we can get - we have/had (it's changing) lots of community testing and reporting and we have very accurate death statistics.
There's also the political complications of reporting - do you bundle it up and report once a week? month? year? or do you report daily? How do you report the numbers? - that person had a heart attack and died, but they also had influenza or COVID. We know COVID can result in cardio-vascular problems, but I don't think any such association has been shown with influenza, so do you report a COVID death with heart attack as dying of COVID, but not a similar case with influenza?
These are only a small sample of the problems with such numbers and such comparisons, so bear that in mind when you look at the data and publications reporting deaths and comparing apples and pears (4).
Edited to add - what do you mean by influenza? Influenza consists of several different viruses from the types A-D, only some of which naturally infect humans and some are more pathogenic than others. A, B and C have been known to infect humans but only A and B types infect humans in significant numbers, and B is almost exclusively a human virus (also infects seals). However, within A we range from highly pathogenic H1N1 (e.g. 1918 influenza pandemic) that transmits among humans well to also highly pathogenic H5N1 that poorly transmits to humans from birds and rarely transmits human:human (but kills >33% of people infected) to low pathogenic H5N1 that also poorly transmits, to H3N2 (one of the seasonal types) which transmits well, but relatively rarely kills people (this is probably the type you are referring to), and finally we have a bunch of influenza A viruses that don't infect humans at all (as far as we can tell; such as these ones). More complexity...
Erikstrup C, Laksafoss AD, Gladov J, Kaspersen KA, Mikkelsen S, Hindhede L, Boldsen JK, Jørgensen SW, Ethelberg S, Holm DK, Bruun MT, Nissen J, Schwinn M, Brodersen T, Mikkelsen C, Sækmose SG, Sørensen E, Harritshøj LH, Aagaard B, Dinh KM, Busch MP, Jørgensen CS, Krause TG, Ullum H, Ostrowski SR, Espenhain L, Pedersen OBV. Seroprevalence and infection fatality rate of the SARS-CoV-2 Omicron variant in Denmark: A nationwide serosurveillance study. Lancet Reg Health Eur. 2022 Oct;21:100479. doi: https://doi.org/10.1016/j.lanepe.2022.100479. Epub 2022 Aug 5. PMID: 35959415; PMCID: PMC9355516
Sigal, A., Milo, R. & Jassat, W. Estimating disease severity of Omicron and Delta SARS-CoV-2 infections. Nat Rev Immunol 22, 267–269 (2022). https://doi.org/10.1038/s41577-022-00720-5
Vanessa Cozza, Harry Campbell, Howard H Chang, A Danielle Iuliano, John Paget, Neha N Patel, Robert C Reiner, Chris Troeger, Cecile Viboud, Joseph S Bresee, Julia Fitzner, Global Seasonal Influenza Mortality Estimates: A Comparison of 3 Different Approaches, American Journal of Epidemiology, Volume 190, Issue 5, May 2021, Pages 718–727, https://doi.org/10.1093/aje/kwaa196
Papadakis, M. (2022). Why comparing coronavirus disease 2019 (COVID-19) and seasonal influenza fatality rates is like comparing apples to pears. Infection Control & Hospital Epidemiology, 43(7), 958-959. doi:https://doi.org/10.1017/ice.2021.140